Solved Exercises on Measuring Perimeters of Shapes in Grade 3

Master perimeter calculations: rectangles, squares, triangles, and irregular polygons through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Rectangle Perimeter
Exercise 1
Find the perimeter of a rectangle with length 7 cm and width 4 cm.
Definition:

Perimeter: The total distance around the outside of a shape

Method for rectangle perimeter:
  1. Add all four sides together
  2. Or use the formula: Perimeter = 2 × (length + width)
  3. Remember: Opposite sides of a rectangle are equal
Length
7 cm
Width
4 cm
Add Sides
7 + 4 + 7 + 4 = 22
Formula
2(7 + 4) = 22
Side Length
Top 7 cm
Right 4 cm
Bottom 7 cm
Left 4 cm
Total 22 cm
Step 1: Identify the dimensions

Length = 7 cm, Width = 4 cm

Rectangle has 2 pairs of equal sides

Step 2: List all side lengths

Top side = 7 cm

Right side = 4 cm

Bottom side = 7 cm

Left side = 4 cm

Step 3: Add all sides

Perimeter = Top + Right + Bottom + Left

Perimeter = 7 + 4 + 7 + 4 = 22 cm

Step 4: Verify with formula

Rectangle perimeter formula: P = 2 × (length + width)

P = 2 × (7 + 4) = 2 × 11 = 22 cm

Perimeter = 22 cm
Final answer:

The perimeter of the rectangle is 22 cm

Applied rules:

• Perimeter definition: Sum of all side lengths

• Rectangle property: Opposite sides are equal in length

• Formula shortcut: P = 2(length + width) for rectangles

Tip 1: Count all sides and add them together.
Tip 2: For rectangles, use the formula P = 2(l + w) to save time.
2 Square Perimeter
Exercise 2
Find the perimeter of a square with side length 5 inches.
Definition:

Square: A rectangle with all four sides equal in length

Side Length
5 inches
Number of Sides
4
Add Sides
5 + 5 + 5 + 5 = 20
Formula
4 × 5 = 20
Side Length
Side 1 5 in
Side 2 5 in
Side 3 5 in
Side 4 5 in
Total 20 in
Step 1: Identify the property of a square

All four sides of a square are equal in length

Each side = 5 inches

Step 2: Add all four sides

Perimeter = Side 1 + Side 2 + Side 3 + Side 4

Perimeter = 5 + 5 + 5 + 5 = 20 inches

Step 3: Use the square formula

Since all sides are equal, multiply side length by 4

Perimeter = 4 × side length

Perimeter = 4 × 5 = 20 inches

Step 4: Verify the calculation

5 + 5 + 5 + 5 = 20 ✓

4 × 5 = 20 ✓

Perimeter = 20 inches
Final answer:

The perimeter of the square is 20 inches

Applied rules:

• Square property: All four sides are equal

• Square formula: Perimeter = 4 × side length

• Multiplication shortcut: Instead of adding four equal numbers, multiply by 4

Tip 1: For squares, multiply the side length by 4.
Tip 2: Remember that all sides of a square are the same length.
3 Triangle Perimeter
Exercise 3
Find the perimeter of a triangle with sides measuring 6 cm, 8 cm, and 10 cm.
Definition:

Triangle: A polygon with three sides and three angles

Side 1
6 cm
Side 2
8 cm
Side 3
10 cm
Perimeter
6 + 8 + 10 = 24 cm
Side Length
Side A 6 cm
Side B 8 cm
Side C 10 cm
Total 24 cm
Step 1: Identify all side lengths

Triangle has three sides with different lengths: 6 cm, 8 cm, and 10 cm

Step 2: Add all three sides together

Perimeter = Side A + Side B + Side C

Perimeter = 6 + 8 + 10 = 24 cm

Step 3: Verify the calculation

6 + 8 = 14, then 14 + 10 = 24 ✓

Perimeter = 24 cm
Final answer:

The perimeter of the triangle is 24 cm

Applied rules:

• Triangle perimeter: Sum of all three side lengths

• General polygon rule: Perimeter = sum of all side lengths

• No special formula: Unlike rectangles/squares, triangles require adding all sides individually

Tip 1: For any polygon, add all side lengths to find the perimeter.
Tip 2: Triangles don't have special formulas like rectangles or squares.
Rules and methods, laws,...
\(\text{Perimeter} = \text{Side}_1 + \text{Side}_2 + \ldots + \text{Side}_n\)
General Polygon Perimeter
\(\text{Rectangle Perimeter} = 2 \times (\text{Length} + \text{Width})\)
Rectangle Formula
Shape 1
\(\text{Square}\)
Perimeter = 4 × side
Shape 2
\(\text{Rectangle}\)
Perimeter = 2(l + w)
Shape 3
\(\text{Triangle}\)
Perimeter = a + b + c
Key definitions:

Perimeter: The total distance around the outside edge of a shape

Polygon: A closed shape with straight sides

Rectangle: A four-sided shape with four right angles and opposite sides equal

Square: A rectangle with all four sides equal in length

Triangle: A three-sided polygon

Side length: The measurement of one side of a shape

Distance: How far it is around or across something

Measure: To find the size or amount of something

Complete methodology for finding perimeter:
  1. Step 1: Identify the shape - Determine if it's a rectangle, square, triangle, or other polygon
  2. Step 2: Measure or identify side lengths - Find the length of each side
  3. Step 3: Choose the method - Use general addition or a specific formula
  4. Step 4: Calculate - Add all side lengths together
  5. Step 5: Check - Verify that all sides were included
  6. Step 6: State the answer - Include the unit of measurement
Tip 1: Always include units in your answer (cm, inches, feet, etc.).
Tip 2: For rectangles, use the formula P = 2(l + w) to save time.
Tip 3: For squares, multiply the side length by 4.
Tip 4: Count all sides before starting to add them.
Common errors: Forgetting to add all sides, using the wrong formula, not including units, miscounting the number of sides.
Memory aid: "Perimeter means go around the edge" - add all the sides.
Essential rules:

• General rule: Perimeter = sum of all side lengths

• Rectangle rule: P = 2 × (length + width)

• Square rule: P = 4 × side length

• Triangle rule: P = side₁ + side₂ + side₃

Solution: Exercises 4 to 5
4 Irregular Polygon
Exercise 4
Find the perimeter of a pentagon with sides measuring 3 cm, 4 cm, 5 cm, 4 cm, and 6 cm.
Definition:

Irregular polygon: A polygon with sides of different lengths

Side Lengths
3, 4, 5, 4, 6 cm
Add All Sides
3 + 4 + 5 + 4 + 6
Calculate
22 cm
Side Length
Side 1 3 cm
Side 2 4 cm
Side 3 5 cm
Side 4 4 cm
Side 5 6 cm
Total 22 cm
Step 1: List all side lengths

The pentagon has 5 sides with lengths: 3 cm, 4 cm, 5 cm, 4 cm, 6 cm

Step 2: Add all side lengths together

Perimeter = 3 + 4 + 5 + 4 + 6

Let's add step by step: 3 + 4 = 7, then 7 + 5 = 12, then 12 + 4 = 16, then 16 + 6 = 22

Step 3: State the answer

Perimeter = 22 cm

Perimeter = 22 cm
Final answer:

The perimeter of the pentagon is 22 cm

Applied rules:

• Irregular polygon rule: Add all side lengths regardless of shape

• General perimeter rule: Sum of all side lengths

• Counting principle: Ensure all sides are included in the sum

Tip 1: For irregular shapes, you must add each side individually.
Tip 2: Double-check that you've included all sides in your calculation.
5 Real-World Application
Exercise 5
A rectangular garden has a length of 12 meters and a width of 8 meters. How much fencing is needed to go around the garden?
Definition:

Real-world application: Using perimeter to solve practical problems like fencing, borders, or frames

Dimensions
Length = 12m, Width = 8m
Formula
P = 2(l + w)
Substitute
P = 2(12 + 8)
Calculate
P = 2(20) = 40m
Side Length
Length 12 m
Width 8 m
Opposite Length 12 m
Opposite Width 8 m
Total Fencing 40 m
Step 1: Identify what the problem is asking

The problem asks for the amount of fencing needed to go around the garden

This is asking for the perimeter of the rectangular garden

Step 2: Identify the dimensions

Length = 12 meters

Width = 8 meters

Step 3: Apply the rectangle perimeter formula

Perimeter = 2 × (length + width)

Perimeter = 2 × (12 + 8) = 2 × 20 = 40 meters

Step 4: State the answer in context

The garden needs 40 meters of fencing to go around it

40 meters of fencing needed
Final answer:

The garden needs 40 meters of fencing

Applied rules:

• Real-world context: Apply geometric formulas to practical problems

• Perimeter application: Perimeter measures the distance around a shape

• Problem interpretation: "Going around" indicates perimeter calculation

Tip 1: Look for words like "around," "fence," "border," or "frame" to identify perimeter problems.
Tip 2: Always include units in real-world problems (meters, feet, etc.).
Key Concepts, Laws, Methods, and Definitions for Perimeter of Polygons
\(\text{Perimeter} = \text{Side}_1 + \text{Side}_2 + \ldots + \text{Side}_n\)
General Perimeter Formula
\(\text{Rectangle Perimeter} = 2 \times (\text{Length} + \text{Width})\)
Rectangle Perimeter Formula
Key definitions:

Perimeter: The total distance around the outside edge of a shape

Polygon: A closed shape made up of straight line segments

Rectangle: A four-sided polygon with four right angles and opposite sides equal

Square: A rectangle with all four sides equal in length

Triangle: A three-sided polygon

Side length: The measurement of one side of a shape

Regular polygon: A polygon with all sides and angles equal

Irregular polygon: A polygon with sides and/or angles of different measures

Distance: How far it is from one point to another

Measure: To find the size, amount, or degree of something

Complete methodology for finding perimeter:
  1. Step 1: Identify the shape - Determine if it's a rectangle, square, triangle, or other polygon
  2. Step 2: List all side lengths - Find the length of each side
  3. Step 3: Choose the appropriate method - Use general addition or a specific formula
  4. Step 4: Calculate - Add all side lengths together
  5. Step 5: Check - Verify that all sides were included and the calculation is correct
  6. Step 6: State the answer - Include the unit of measurement
Tip 1: Always include units in your answer (cm, inches, feet, meters).
Tip 2: For rectangles, use the formula P = 2(l + w) to save time.
Tip 3: For squares, multiply the side length by 4.
Tip 4: Count all sides before starting to add them.
Tip 5: Look for key words in word problems: "around," "fence," "border," "frame," "perimeter."
Common errors to avoid: Forgetting to add all sides, using the wrong formula, not including units, miscounting the number of sides, confusing perimeter with area.
Memory aid: "Perimeter means go around the edge" - add all the sides.
Essential formulas and rules:

• General rule: Perimeter = sum of all side lengths

• Rectangle rule: P = 2 × (length + width)

• Square rule: P = 4 × side length

• Triangle rule: P = side₁ + side₂ + side₃

• Regular polygon rule: P = number of sides × length of one side

Detailed Summary: Perimeter of Polygons

Definitions of Key Concepts:
Perimeter: The total distance around the outside edge of a shape (e.g., for a rectangle with sides 3 and 4, perimeter = 3+4+3+4 = 14).
Polygon: A closed shape made up of straight line segments (triangle, square, rectangle, pentagon, etc.).
Rectangle: A four-sided shape with four right angles and opposite sides equal in length.
Square: A rectangle with all four sides equal in length.
Triangle: A three-sided polygon.
Side length: The measurement of one side of a shape.
Regular polygon: A polygon with all sides and angles equal (e.g., equilateral triangle, square).
Irregular polygon: A polygon with sides and/or angles of different measures.
Distance: How far it is from one point to another.
Measure: To find the size, amount, or degree of something.
Core Rules and Principles:
General Perimeter Rule: Perimeter = sum of all side lengths (works for any polygon).
Rectangle Rule: P = 2 × (length + width) (since opposite sides are equal).
Square Rule: P = 4 × side length (since all four sides are equal).
Triangle Rule: P = side₁ + side₂ + side₃ (add all three sides).
Regular Polygon Rule: P = number of sides × length of one side.
Perimeter Property: The perimeter is always measured in linear units (cm, inches, feet, meters).
Step-by-Step Methods and Procedures:
General Perimeter Method: 1) Identify all sides of the polygon, 2) Measure or note the length of each side, 3) Add all side lengths together, 4) Include the unit of measurement in your answer.
Rectangle Perimeter Method: 1) Identify length and width, 2) Use formula P = 2(l + w), 3) Calculate the sum inside parentheses first, 4) Multiply by 2.
Square Perimeter Method: 1) Identify the side length, 2) Use formula P = 4s, 3) Multiply side length by 4.
Real-World Application Method: 1) Read the problem carefully, 2) Identify what is being asked (perimeter), 3) Find the dimensions, 4) Apply the appropriate formula, 5) State the answer with units.
Concrete Examples:
Simple Example: Square with side length 5: P = 4 × 5 = 20 units.
Intermediate Example: Rectangle with length 7 and width 4: P = 2(7 + 4) = 2(11) = 22 units.
Advanced Example: Irregular pentagon with sides 3, 4, 5, 4, 6: P = 3 + 4 + 5 + 4 + 6 = 22 units.
Tips, Tricks, and Common Pitfalls:
Tip 1: Always include units in your answer (cm, inches, feet, meters).
Tip 2: For rectangles, use the formula P = 2(l + w) to save time.
Tip 3: For squares, multiply the side length by 4.
Tip 4: Count all sides before starting to add them.
Tip 5: Look for key words in word problems: "around," "fence," "border," "frame," "perimeter."
Pitfall 1: Forgetting to add all sides of the polygon.
Pitfall 2: Using the wrong formula for the shape.
Pitfall 3: Not including units in the final answer.
Pitfall 4: Confusing perimeter with area.
Key Notes for Memorization:
Mnemonic: "Perimeter means go around the edge" - add all the sides.
Mnemonic: "Peri-meter: around-measure" - it measures around the outside.
Key Point: Perimeter is always measured in linear units (not square units like area).
Key Point: For rectangles: P = 2(l + w); for squares: P = 4s; for triangles: P = a + b + c.
Explanation for Students:
Perimeter is like walking around the outside of a shape! If you wanted to put a fence around your yard, measure the distance around your room, or put a frame around a picture, you'd be finding the perimeter. For rectangles, you can use the special formula P = 2(l + w) because opposite sides are equal. For squares, since all sides are the same, just multiply the side length by 4. For triangles and other shapes, just add up all the side lengths. Remember to always include the unit of measurement (like cm, inches, or feet) in your answer!